1. What is 2's Complement?

2's complement is a mathematical operation used in computing to represent signed integers in binary form. It's the most common method for representing signed numbers in digital systems and computers.

2. How Does 2's Complement Work?

The 2's complement is calculated using the formula:

Where:

• \( \sim Decimal \) — Bitwise NOT of the decimal number
• \( + 1 \) — Add 1 to the result

Explanation: For positive numbers, 2's complement is the same as the binary representation. For negative numbers, it's calculated by inverting all bits of the absolute value and adding 1.

3. Importance of 2's Complement

Details: 2's complement representation allows for simple binary arithmetic operations (addition and subtraction) using the same hardware, eliminates the problem of negative zero, and provides a symmetric range of representable numbers.

4. Using the Calculator

Tips: Enter a decimal integer and select the number of bits for representation. The calculator will show the 2's complement binary representation. Make sure the number fits within the selected bit range.

5. Frequently Asked Questions (FAQ)

Q1: Why use 2's complement instead of sign-magnitude?
A: 2's complement simplifies arithmetic operations and eliminates the negative zero problem found in sign-magnitude representation.

Q2: What is the range of numbers for n-bit 2's complement?
A: The range is from \(-2^{n-1}\) to \(2^{n-1}-1\). For example, 8-bit range is -128 to 127.

Q3: How is subtraction performed using 2's complement?
A: Subtraction A - B is performed as A + (-B), where -B is the 2's complement of B.

Q4: What happens if a number exceeds the bit range?
A: Overflow occurs, and the result will be incorrect due to wrapping around the maximum representable value.

Q5: Can 2's complement represent zero?
A: Yes, 2's complement has a single representation for zero (all bits 0), unlike sign-magnitude which has positive and negative zero.